Let zk=e^2π i k/7 for k=1,2,…,6 be the non-real seventh roots of unity.

The $z_k$ are roots of $1+z+\dots+z^6=0$, so $\sum z_k=-1$, $\sum z_k^2=-1$ and, since $\prod(x-z_k)=1+x+\dots+x^6$, putting $x=1$ gives $\prod(1-z_k)=7$, so (B) is true. For (A): $|1-z_k|^2=2-2\cos\frac{2\pi k}{7}$, so the sum is $12-2(-1)=14$, true. For (C): $\frac{1}{1-z_k}$ has real part $\frac12$ and the imaginary parts cancel in conjugate pairs, so the sum is $6\cdot\frac12=3$, not $\frac72$; false. For (D): $\sum(1-2z_k+z_k^2)=6+2-1=7$, true. Each statement uses a different symmetric function of the roots of unity, and none needs the explicit angles beyond the cosine sum.